The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.126.8. Suppose that $R = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } R_\lambda $ is a directed colimit of rings. Then the category of finitely presented $R$-algebras is the colimit of the categories of finitely presented $R_\lambda $-algebras. More precisely

  1. Given a finitely presented $R$-algebra $A$ there exists a $\lambda \in \Lambda $ and a finitely presented $R_\lambda $-algebra $A_\lambda $ such that $A \cong A_\lambda \otimes _{R_\lambda } R$.

  2. Given a $\lambda \in \Lambda $, finitely presented $R_\lambda $-algebras $A_\lambda , B_\lambda $, and an $R$-algebra map $\varphi : A_\lambda \otimes _{R_\lambda } R \to B_\lambda \otimes _{R_\lambda } R$, then there exists a $\mu \geq \lambda $ and an $R_\mu $-algebra map $\varphi _\mu : A_\lambda \otimes _{R_\lambda } R_\mu \to B_\lambda \otimes _{R_\lambda } R_\mu $ such that $\varphi = \varphi _\mu \otimes 1_ R$.

  3. Given a $\lambda \in \Lambda $, finitely presented $R_\lambda $-algebras $A_\lambda , B_\lambda $, and $R_\lambda $-algebra maps $\varphi _\lambda , \psi _\lambda : A_\lambda \to B_\lambda $ such that $\varphi \otimes 1_ R = \psi \otimes 1_ R$, then $\varphi \otimes 1_{R_\mu } = \psi \otimes 1_{R_\mu }$ for some $\mu \geq \lambda $.

Proof. To prove (1) choose a presentation $A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$. We can choose a $\lambda \in \Lambda $ and elements $f_{\lambda , j} \in R_\lambda [x_1, \ldots , x_ n]$ mapping to $f_ j \in R[x_1, \ldots , x_ n]$. Then we simply let $A_\lambda = R_\lambda [x_1, \ldots , x_ n]/(f_{\lambda , 1}, \ldots , f_{\lambda , m})$.

Parts (2) and (3) follow from Lemma 10.126.7. $\square$


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